PROBABILTY
Q1.
A basket contains 6 red, 5 green and 8 blue balls. If
four balls are picked at random, what is the
probability that all four of them are either red or any
two out of the four are green ?
(a) 5/1292
(b) 925/3876
(c) 359/1938
(d) 11/3876
(e) None of these
Q2.
Directions : Study the given information carefully to
answer the questions that follow . A basket contains 6
blue, 2 red, 4 green and 3 yellow balls. If 2 balls are
picked at random, what is the probability that either
both are green or both are yellow?
(a) 2/5
(b) 3/35
(c) 1/3
(d) 3/91
(e) None of these
Q3.
If 5 balls are picked at random, what is the probability
that at least one is blue ?
(a) 137/143
(b) 9/91
(c) 18/455
(d) 2/5
(e) None of these
Q4.
If 2 balls are picked at random, what is the probability
that both are blue ?
(a) 1/5
(b) 8/91
(c) 2/15
(d) 7/27
(e) None of these
Q5.
If 4 balls are picked at random, what is the probability
that 2 are red and 2 are green?
(a) 4/15
(b) 5/27
(c) 1/3
(d) 2/455
(e) None of these
Q6.
If 3 balls are picked at random, what is the
probability that none is yellow?
(a) 3/455
(b) 1/5
(c) 44/91
(d) 4/5
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(e) None of these
Q7.
A basket contains three blue and four red balls. If
three balls are drawn at random from the basket,
what Is the probability that till the three are either
blue or red ?
(a)1
(b) 1/7
(c) 3/14
(d) 3/28
(e) None of these
Q8.
Direction : Study the given information carefully to
answer the questions that follow. An urn contains 4
green, 5 blue, 2 red and 3 yellow marbles. If two
marbles are drawn, at random, what is the probability
that both are red Or at least one is red ?
(a) 26/93
(b) 2/7
(c) 199/366
(d) 135/193
(e) None of these
Q9.
If three marbles are drawn at random, what is the
probability that at least one is yellow?
(a) 1/5
(b) 199/364
(c) 125/364
(d) 5/9
(e) None of these
Q10.
If eight marbles are drawn at random, what is the
probability that there are equal number of marbles of
each colour?
(a) 3/8
(b) 351/738
(c) 60/1001
(d) 1/1001
(e) None of these
Q11.
If three marbles are drawn at random, what is the
probability that none is green ?
(a) 3/8
(b) 273/748
(c) 30/91
(d) 41/91
(e) None of these
Q12.
If four marbles are drawn at random, what is the
probability that two are blue and two are red ?
(a) 10/1001
(b) 7/17
(c) 15/384
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(d) 3/5
(e) None of these
Q13.
A bag contains 13 white and 7 black balls. Two balls
are drawn at random. What is the probability that
they are of the same colour?
(a) 41/190
(b) 41/190
(c) 59/190
(d) 99/190
(e) 77/190
Q14.
From a well-shuffled pack of 52 playing cards, one
card is drawn at random. What is the probability that
the card drawn will be a black king ?
(a) 1/26
(b) 7/13
(c) 3/13
(d) 9/13
(e) 1/13
Q15.
Directions : Read the following information to answer
these questions. There are 9 red, 7 white and 4 black
balls in an urn. Two balls are drawn at random from
the um. What will be the probability that both the
balls are red ?
(a) 18/95
(b) 9/95
(c) 9/10
(d) 8/95
(e) 5/19
Q16.
If two balls are selected at random what is the
probability that one ball is white and the other ball is
red ?
(a) 91/190
(b) 63/190
(c) 5/19
(d) 4/95
(e) 71/190
Q17.
A bag contains 3 red balls, 5 yellow balls and 7 pink
balls. If one ball is drawn at random from the bag,
what is the probability that it is either pink or red ?
(a) 1/3
(b) 2/3
(c) 1/4
(d) 2/5
(e) None of these
Q18.
A bag contains 5 red balls, 7 yellow balls and 3 pink
balls. If two balls are drawn at random from the bag,
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one after another, what is the probability that the first
ball is red and the second ball is yellow?
(a) 5/12
(b) 3/8
(c) 1/4
(d) 1/8
(e) 1/6
Q19.
There are 6 red balls, 5 yellow and 3 pink balls in an
um. Two balls are drawn at random. What Is the
probability that none of the drawn balls is of red
colour ?
(a) 8/13
(b) 7/13
(c) 6/13
(d) 5/13
(e) 4/13
Q20.
A bag contains 4 red, 5 yellow and 6 pink balls. Two
balls are drawn at random. What is the probability
that none of the balls drawn are yellow in colour?
(a)(1)1/7
(b)(2)3/7
(c)(3)2/7
(d)(4)5/14
(e)(5)9/14
Q21.
A bag contains 6 black and 8 white balls. One ball is
drawn at random. What is the probability that the ball
drawn Is white?30) 4
(a) 3/4
(b) 4/7
(c) 1/8
(d) 3/7
(e) 1/4
Q22.
What is the probability that a number selected from
numbers 1, 2, 3 30, is prime number, when each of the
given numbers is equally likely to be selected?
(a) 9/30
(b) 8/30
(c) 10/30
(d) 11/30
(e) 21/30
Q23.
An urn contains 9 red , 7 white and 4 black balls . If
two balls are drown at random then find the
probability that both the balls are random ?
(a) 17/95
(b) 18/95
(c) 1/12
(d) 91/190
(e) None of these
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Q24.
Out of 5 girls and 3 boys, 4 children are to be
randomly selected for a quiz contest. What is the
probability that all are girls ?
(a) 1/14
(b) 1/7
(c) 5/17
(d) 2/17
(e) None of these
Q25.
Directions : Study the given information carefully and
answer the questions that follow: A basket contains 4
red, 5 blue and 3 green marbles. If three marbles are
picked at random, what is the probability that either
all are green or all are red ?
(a) 7/44
(b) 7/12
(c) 5/12
(d) 1/44
(e) None of these
Q26.
If two marbles are picked at random, what is the
probability that both are red ?
(a) 3/7
(b) 1/2
(c) 2/11
(d) 1/6
(e) None of these
Q27.
If three marbles are picked at random, what is the
probability that at least one is blue ?
(a) 7/12
(b) 37/44
(c) 5/12
(d) 7/44
(e) None of these
Q28.
Directions :Study the given Information carefully and
answer the questions that follow: An urn contains 6
red, 4 blue, 2 green and 3 yellow marbles. If four
marbles are picked at random, what is the probability
that at least one is blue
(a) 4/15
(b) 69/91
(c) 11/15
(d) 22/91
(e) None of these
Q29.
If two marbles are picked at random, what is the
probability that both are red ?
(a) 1/6
(b) 1/3
(c) 2/15
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(d) 2/5
(e) None of these
Q30.
If three marbles are picked at random, what is the
probability that two are blue and one is yellow?
(a) 3/91
(b) 1/5
(c) 18/455
(d) 7/15
(e) None of these
Q31.
If four marbles are picked at random, what is the
probability that one is green, two are blue and one is
red ?
(a) 24/455
(b) 13/35
(c) 11/15
(d) 1/3
(e) None of these
Q32.
If two marbles are picked at random, what is the
probability that either both are green or both are
yellow ?
(a) 5/91
(b) 1/35
(c) 1/3
(d) 4/105
(e) none of these
Q33.
A bag contains 9 white and 5 blackballs. Two balls are
drawn at random. What is the probability that they
are of the same colour ?
(a) 47/91
(b) 46/91
(c) 45/91
(d) 2/3
(e) None of these
Q34.
A dice is thrown twice. What is the probability of
getting a sum 9 from both the throws ?
(a) 1/9
(b) 2/9
(c) 1/3
(d) 3/4
(e) None of these
Q35.
Three balls are drawn at random from the bag What
is the probability that 2 balls will be red and 1 ball will
be green?
(a) 5/68
(b) 3/ 68
(c) 7/68
(d) 9/ 68
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(e) None of these
Q36.
What is the probability of drawing one ball that will
be neither red nor green ?
(a) 1/3
(b) 4/9
(c) 5/9
(d) 2/3
(e) None of these
Q37.
What is the probability of drawing two balls of same
colour ?
(a) 13/153
(b) 48/153
(c) 49/153
(d) 46/153
(e) None of these
Q38.
Directions : Study the information carefully to answer
these questions. A bag contains 7 blue balls and 5
yellow balls. If two balls are selected at random, what
is the probability that none is yellow ?
(a) 5/33
(b) 5/22
(c) 7/22
(d) 7/33
(e) 7/66
Q39.
A die is thrown twice. What is the probability of
getting a sum 7 from both the throws ?
(a) 5/18
(b) 1/18
(c) 1/9
(d) 1/6
(e) none of these
Q40.
What is the difference between the number of females
who are non-graduates and the number of males who
are graduates ?
(a)2
(b)24
(c)4
(d)116
(e)36
Q41.
What is the sum of the number of females who are
graduates and the number of males who are non
graduates ?
(a)184
(b)96
(c)156
(d)84
(e)196
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Q42.
What is the ratio between the total number of males
and the number of females who are non-graduates ?
(a) 6:1
(b) 8 :1
(c) 8 : 3
(d) 5 :2
(e) 7:2
Q43.
Directions : Study the information carefully to answer
these questions. In a bag there are 6 red balls. 4 green
balls and 8 yellow balls Three balls are drawn at
random from the bag. What is the probability that 2
balls will be red and 1 ball will be green?
(a) 5/68
(b) 3/68
(c) 7/ 68
(d) 9/68
(e) None of these
Q44.
What is the probability of drawing one ball that will
be neither red nor green ?
(a) 1/3
(b) 4/9
(c) 5/9
(d) 2/3
(e) none of these
Q45.
what is the probability of drawing to balls at the same
?
(a) 53/153
(b) 48/153
(c) 49/153
(d) 46/153
(e) none of these
Q46.
Directions : Study the information carefully to answer
these questions. A bag contains four blue shirts, five
red shirts and six yellow shirts. Three shirts are
drawn randomly. What is the probability that exactly
one of them is blue ?
(a) 36/91
(b) 40/91
(c) 44/91
(d) 48/91
(e) 31/91
Q47.
One shirt is drawn randomly. What is the probability
that it is either red or yellow ?
(a) 4/15
(b) 7/15
(c) 11/15
(d) 8/15
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(e) 13/15
Q48.
Two shirts are is drawn randomly. What is the
probability that both of them are blue ?
(a) 3/35
(b) 1/35
(c) 2/35
(d) 4/35
(e) 6/35
Q49.
In a bag there are 4 white, 4 red and 2 green balls.
Two balls are drawn at random. What is the
probability that at least one ball is of green colour ?
(a) 5/7
(b) 10/21
(c) 2/7
(d) 11/21
(e) None of these
Q50.
A bag contains 5 red balls, 6 yellow and 3 green balls.
If two balls are picked at random, what is the
probability that both are red or both are green in
colour ?
(a) 2/15
(b) 1/15
(c) 2/7
(d)1
(e) none of these
Q51.
A bag contains 4 red balls, 6 green balls and 5 blue
balls. If three balls are picked at random, what is the
probability that two of them are green and one of
them is blue in colour ?
(a) 20/21
(b) 10/91
(c) 15/91
(d) 5/91
(e) 25/91
Q52.
Directions : Study the information carefully to answer
these questions. A bag contains 2 red, 3 green and 2
blue balls. 2 balls are to be drawn randomly. What is
the probability that the balls drawn contain no blue
ball ?
(a)(1)5/7
(b)(2)10/21
(c)(3) 2/7
(d)(4)11/21
(e)(5) none of these
Q53.
If two marbles are picked at random, what is the
probability that both are green ?
(a) 2/l5
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(b) 1/l5
(c) 2/7
(d)1
(e) None of these
Q54.
If three marbles are picked at random, what is the
probability that two are blue and one is yellow ?
(a) 2/15
(b) 6/91
(c) 12/91
(d) 3/15
(e) None of these
Q55.
If four marbles are picked at random, what is the
probability that at least one is yellow ?
(a) 91/123
(b) 69/91
(c) 125/143
(d) 1/4
(e) None of these
Q56.
If two marbles are picked at random, what is the
probability that either both are red or both are green
?
(a) 3/5
(b) 4/105
(c) 2/7
(d) 5/91
(e) None of these
Q57.
If four marbles are picked at random, what is the
probability that one is green, two are blue and one is
red ?
(a) 4/15
(b) 17/280
(c) 6/91
(d) 11/15
(e) None of these
Q58.
There are 8 brown balls, 4 orange balls and 5
blackballs in a bag. Five balls are chosen at random,
What is the probability of their being 2 brown balls, 1
orange ball and 2 black balls ?
(a) 191/1547
(b) 180/1547
(c) 280/1547
(d) 189/1547
(e) NONE OF THESE
Q59.
A bag A contains 4 green and 6 red balls. Another bag
B contains 3 green and 4 red balls. If one ball is drawn
from each bag, find the probability that both are green
(a) 13/70
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(b) 1/4
(c) 6/35
(d) 8/35
(e) None of these
Q60.
A box contains 14 eggs out of which 8 are rotten. Two
eggs are chosen at random. What is the probability
that none of the chosen eggs is rotten ?
(a) 5/23
(b) 10/23
(c) 12/91
(d) 15/91
(e) 8/91
Q61.
A bag A contains 4 green and 6 red balls. Another bag
B contains 3 green and 4 fed balls. If one ball Is drawn
from each bag, find the probability that both are
green.
(a) 13/70
(b) 1/4
(c) 6/35
(d) 8/35
(e) NONE OF THESE
Q62.
A bag contains 13 white and 7 black balls. Two balls
are drawn at random, What is the probability that
they are of the same colour ?
(a) 41/190
(b) 21/190
(c) 59/190
(d) 99/190
(e) 77/190
Q63.
A box contains 4 black balls, 3 red balls and 5 green
balls 2 balls are drawn from the box at random. What
is the probability that both the balls are of the same
colour ?
(a)47/68
(b) 1/6
(c)19/66
(d) 2/11
(e) None of these
Q64.
Three balls are drawn from the um. What Is the
probability that two balls are red and 1 ball is blue ?
(a) 11/408
(b) 37/408
(c) 35/408
(d) 1/102
(e) None of these
Q65.
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An urn contains 3 red and 4 green marbles. If three
marbles are picked at random, what is the probability
that two are green and one is red ?
(a) 3/7
(b) 18/35
(c) 5/14
(d) 4/21
(e) None of these
Q66.
A property tax increase was proposed by a municipal
corporation and it was observed that 40% of the
property owners favored it while 80% of the nonowners of property favored it. If 70% of voters are
property owners, what is the probability that a voter
selected at random would be the one favoring the
increase?
(a)0.8
(b)0.52
(c)0.4
(d) Data Inadequate
(e) None of these
Q67.
In a container there are 28 eggs out of which 8 eggs
are rotten. If two eggs are chosen at random, what will
be the probability that at least one egg is rotten ?
(a) 94/189
(b) 95/187
(c) 93/189
(d) 97/189
(e) None of these
Q68.
The circle given below is divided into eight sectors of
equal area. What is the probability that the spinner
will land on an even-numbered region in each of two
consecutive spins?
(a) 25/10
(b) 3/8
(c) 1/4
(d) 9/64
(e)
Q69.
A class of 30 students occupy a classroom containing
5 rows of seats, with 8 seats in each row. If the
students seat themselves at random, the probability
that the sixth seat in the fifth row will be empty is :
(a) 1/5
(b) 1/3
(c) 1/4
(d) 2/5
(e)
Q70.
Directions : Read the following information to answer
the questions given below it. An um contain 5 red
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balls, 6 green balls and 7 blue balls. Two balls are
drawn at random from the um. What is the probability
that both the balls are of red colour ?
(a) 10/153
(b) 10/53
(c) 3/31
(d) 2/31
(e) None of these
Q71.
Directions : Read the following information to answer
the questions given below it. An um contain 5 red
balls, 6 green balls and 7 blue balls. Two balls are
drawn at random from the sum Two balls are drawn
at random from the um. What is the probability that
both the balls are of green colour ?
(a) 1/17
(b) 5/51
(c) 2/17
(d) 7/51
(e) 6/51
Q72.
There are 4 red balls, 4 green balls and 6 blue balls in
a box. If two balls are drawn randomly, what is the
probability that at least one of them is blue ?
(a) 6/13
(b) 7/13
(c) 8/13
(d) 10/13
(e) 9/13
Q73.
If one ball is drawn randomly, what is the probability
that it is either red or blue ?
(a) 5/9
(b) 7/9
(c) 4/7
(d) 3/7
(e) 5/7
Q74.
If three balls are drawn randomly, what is the
probability that one of them is green and the other
two are blue?
(a)(1) 15/91
(b)(2) 24/91
(c)(3) 20/91
(d)(4) 10/91
(e)(5)12/91
Q75.
There are 6 red balls and 4 yellow balls in a bag. Two
balls are simultaneously drawn at random. What is
the probability that both the balls are of same colour ?
(a) 7/15
(b) 5/12
(c) 7/11
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(d) 7/8
(e) 1/8
Q76.
There are 7 red balls and 8 yellow balls in a bag. Two
balls are simultaneously drawn at random. What is
the probability that both the balls are of same colour ?
(a) 3/18
(b) 13/30
(c) 3/10
(d) 7/18
(e) 7/15
ANSWERS :
1b
7b
13 d
19 e
25 d
31 a
37 c
43 a
49 d
55 b
61 c
67 a
73 e
2b
8e
14 a
20 b
26 e
32 d
38 c
44 b
50 c
56 b
62 d
68 b
74 d
3a
9b
15 a
21 b
27 b
33 b
39 d
45 c
51 c
57 c
63 c
69 a
75 a
4e
10 c
16 b
22 c
28 b
34 a
40 c
46 c
52 b
58 c
64 c
70 a
76 e
5d
11 c
17 b
23 b
29 e
35 a
41 e
47 c
53 e
59 c
65 b
71 b
6c
12 a
18 e
24 a
30 c
36 b
42 b
48 c
54 c
60 d
66
72 e
1.(2) Number of balls = 5 + 5 + 8 = 19
Exhaustive number of cases = Ways of selecting4 balls
out of 19
= 19C4 = (19 × 18 × 17 × 16)/ (1 × 2 × 3 × 4) = 3876
Favourable number of cases = Selecting 4 red balls or
any two green balls out the four
=6C4 + 5C2 × 14C2
= (6 × 5 × 4 × 3)/(1 × 2 × 3 × 4) + (5 × 4)/2 × (14 ×
13)/2
= 15 + 910 = 925
∴ Required probability = 925/3876
2.(2) Total number of balls in the basket = 15
Exhaustive number of cases = Number of ways of
selecting 2 balls out of 15 balls = 15C2
= (15 × 14)/ (1 × 2) = 105
Favourable number of cases
= 4c2 + 3c2
= (4 × 3)(1 × 2) + (3 × 2)/(1 × 2) = 6 + 3 = 9
∴ Required probability = 9/105 = 3 / 35
3.(1) Exhaustive number of cases = Number of ways of
selecting 5
balls out of 15 balls = 15C5=
Let no blue ball be selected.
∴ Number of ways of selecting 5 balls out of 9 balls with
blue balls = 9C5
Required probability= 1 – 9C5/ 15C5
= 1 - (9 × 8 × 7 × 6 × 5)/(15 × 14 × 13 × 12 × 11)
= 1- 6/143 = 137/143
4.(5) Exhaustive number of cases
= 15C2 = (15 × 14)(1 × 2)
= 105
Favourable number of cases
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= 6C2 = (6× 5)/(1 × 2) = 15
∴ Required probability = 15/105 = 1/7
5.(4) Exhaustive number of cases
= 15C4 = (15 × 14 × 13 × 12)/(1 × 2 × 3 × 4) = 1365
Favourable number of cases
= 2C2 × 4C2 = 1 × (4 × 3)/(1 × 2) = 6
∴ Required probability = 6/1365 = 2/455
6.(3) Exhaustive number of cases
= 15C3 = (15 × 14 × 13)/(1 × 2 × 3) = 455
Favourable number of cases
= 12C3= (12 × 11 × 10)/(1 × 2 × 3) = 220
∴ Required probability
= 220/455 = 44/91
7.(2) Number of possible outcomes= n(S) = 7C3= (7 × 6
×5)/(1 × 2× 3)
Favorable number ofcases
= n (E) = 3C3 + 4C3= 1 + 4 = 5
∴ Required probability = n(E)/n(S)
= 5/35 = 1/7
8.(5) Total number of marbles in the urn
= 4 + 5 + 2 + 3 = 14
Total possible outcomes= selection of2 marbles out of
14 marbles
= 14C2 = (14 × 13)/(1 × 2) = 91
Favourable number of cases
= 2C2 + 2C1 × 12C1
= 1 + 2 × 12 = 25
∴ Required probability = 25/91
9.(2) Total possible outcomes = 14C3
= (14 × 13 × 12)/ (1 × 2 × 3) = 364
When no marble is yellow,
Favourable number of cases
= 11C3 = (11 × 10 × 9)/(1 × 2× 3) = 165
∴ Probability that no marble is yellow = 165/364
∴ Required probability
= 1 – 165/364 = (364 – 165)/364 = 199/364
10.(3) Total possible outcomes
= 14C8 = 14C6
[∴nCr = nCn-r]
=(14 × 13 × 12 ×11 × 10 × 9)/ (1 × 2 × 3 × 4 × 5 × 6) =
3003
Favourable number of cases
= 4C2 × 5C2 × 2C2 × 3C2
= 6 × 10 × 1 × 3 = 180
∴ Required probability
= 180 /3003 = 60/1001
11.(3) Total possible outcomes
= 14C = (14 × 13 × 12)/(1 × 2 × 3) = 364
No ball is green.
∴ Total favourable outcomes = selection of3 marbles
out5 blue, 2 red and 3 yellow marbles.
= 10C3 = (10 × 9 × 8)/(1 × 2 × 3 ) = 120
∴ Required probability = 120/364= 30/91
12.(1) Total possible outcomes = 14C4
= (14 × 13 × 12 × 11)/ (1 × 2 × 3 × 4) = 1001
Favourable outcomes
= 5C2 × 2C2 = 10 × 1 = 10
∴ Required probability = 10/1001
13.(4) Total possible outcomes
= 20C2 = (20 × 19)/(1 × 2) = 190
Total favourable outcomes
= 13C2 + 7C2 = (13× 12)/(1 × 2) + (7 × 6)/(1×2)
= 78 + 21 = 99
∴ Required probability = 99/190
14.(1) Total possible outcomes = 52C1 = 52
Favourable outcomes = 2 = 2C1
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∴ Required probability
= 2/52 = 1/26
15.(1) Total possible outcomes = Selection of 2 balls out
of (9 + 7 + 4) =20 balls
=20C2 = (20 × 19)/(1 × 2) = 190
Favourable outcomes = Selection of 2 balls out of 9 red
balls
=9C2 = (9 × 8)/ (1 × 2) =36
∴ Required probability
= 36/190 =18/95
16.(2) Total possible outcomes = 190
Favourable outcomes = Selectionof1 Ball out of 9 red
balls and 1 ball from 7 white balls
= 9C1 × 7C1 = 9 × 7 = 63
∴ Required probability = 63/190
17.(2) Total number of balls in the bag = 3 + 5 + 7 = 15
One ball is taken out.
∴ Total possible outcomes = 15
Total favourable outcome = selection of1 ball out of10
balls = 10
∴ Required probability
= 10/15 = 2/3
18.(5) Probability of the first ball to be red
19.(5) Total number of balls in the urn = 6 + 5 + 3 = 14
Total possible outcomes
= Selection of 2 balls out of14 balls = 14C2
= 14 × 13/ 1 × 2 = 91
The drawn balls are not of red colour.
Total favourable outcomes = Selection of 2 balls out of
remaining 8 balls
= 8C2 = (8 × 7)/(1 × 2)
∴ Required probability
= 28/91 = 4/13
20.Total number of balls in the bag = 4 + 5 + 6 = 15
Total possible outcomes = Selection of 2 balls out of 15
balls
= 15C2 = (15 × 14)/(1 × 2) = 105
Total favourable outcomes = Selection of 2 balls out of 4
orange and 6 pink balls
= 10C2 = (10 × 9) / (1 × 2) = 45
∴ Required probability
= 45/105 = 3/7
21.(2) Number of balls in the bag = 6 + 8 = 14
Total possible outcomes = Selection of 1 ball out of 14
calls
= 14C1 = 14
Total favourable outcomes
= Selection of 1 balls = 8C1 = 8
∴ Required probability
=8/14 = 4/7
22.(3) Prime numbers in 1, 2, 3, ……. 30.
= 2, 3,5 ,7,11,13,17,19,23,29 = 10
Required probability = 10/30 = 1/3
23.(1) There are 20 balls in the urn out of which 2 ball
cab be drawn in 20C2 ways.
∴ Total number of elementary events = 20C2 = 190
There are 9 red balls out of which 2 ball can be drawn in
9C ways.
2
∴ Favourable number of elementary events = 9C2
= (9 × 8)/(1 × 2) = 36
∴ Required probability = 36/190 = 18/95
24.(1) Total possible outcomes = Selection of 4 children
out of 8 boys and girls
= 8C4 (8 × 7 × 6 × 5)/(1 × 2 × 3 × 4) = 70
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Favourable number of cases = Selection of4girls out of 5
girls
= 5C4 = 5C1 = 5
[nCr = nCn-r]
∴ Required probability
= 5/70 = 1/14
25.(4) Total possible outcomes= Number of ways of
picking 3 marbles out of 12 marbles =n(S)
= 12C3 = (12 × 11 × 10)/(1 × 2 × 3) = 220
Favourable number of cases = n[E]
= 3C3 + 4C3 = 1 + 4 = 5
∴ Required Probability
= n(E)/ n(S) = 5/220 = 1/44
26.(5) Total possible outcomes
= n (S) = 12C2 = 12 × 11/(1 × 2) = 66
Favourable number of cases = n(E)
= 4C2 = (4 × 3)/ (1 × 2) = 6
∴ Required probability
= n(E)/ n(S) = 6/66 = 1/11
27.(2) Total possible outcomes = n(S) = 12C3 = 220
Favourable number of cases n(E)
= Number of ways of picking 3 marbles (none is blue)
out of 7 marbles
= 7C3 = (7 × 6 × 5)/(1 × 2 × 3) = 35
∴ Required probability
= (1 – 35/220) = 1 – 7/44 = 37/44
28.Total possible outcomes= Total possible outcomes =
n(S) = Selection of 4 marbles out of15 marbles.
= 15C4 = (15 × 14 × 13 × 12)/(1 × 2 ×3 ×4) = 1365
When no marble is blue, favourable number of cases
n(E)
= Selection of 4 marbles out of 11 marbles
= 11C4 = (11 × 10 × 9 × 8)/(1 × 2 × 3 × 4) = 330
∴ Required probability = 1 – n(E)/n(S)
= 1 – 330 / 1365 = 1 – 22/ 91 = 69/91
29.(5) Total possible outcomes = n(S)
= 15C2 = (15 × 14) / (1 × 2) = 105
Favourable number of cases
= n(E) Selection of 2 marbles out of 6 red marbles
= 6C2 = (6 × 5)/(1 × 2) = 15
∴ Required probability
= n(E)/n(S) = 15/105 = 1/7
30.(3) Total possible outcomes = n(S)
= 15C3= (15 × 14 × 13)/ (1 × 2 ×3) = 455
Favourable no. of cases
= n (E) = 4C2 × 3C1
= (4 ×3)/ (1 × 2) × 3 = 18
∴ Required probability = 18/455
31.(1) Total possible outcomes = n (S)
= 15C4 = 1365
Favourable number of cases = 2C1 × 4C2× 6C1
2 × (4 × 3)/ (1 × 2) × 6 = 72
∴ Required probability = 72/1365 = 24/455
32.(4) Total possible outcomes= n(S) = 15C2 = 105
Favourable number of cases
= n (E) = 2C2 + 3C2 = 1 + 3 = 4
∴ Required probability = 4/105
33.(2) Total number of balls in the bag = 9 + 5 = 14
Two balls are drawn
∴ Total possible outcomes
= 14C2 = (14 × 13)/(1 × 2) = 91
Both the balls should be either white or black.
∴ Favourable outcomes
= 9C2 + 5C2 = (9 ×8)/(1 × 2) + (5 × 4)/(1 × 2)
= 36 + 10 = 46
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∴ Required probability = 46/91
34.(1) Dice is thrown twice.
∴ Total possible outcomes = 6 × 6 = 36
Pairs whose sum is 9 = (3 , 6) (6 , 3) (4 , 5) and(5 , 4)
∴ Favourable probability = 4/36 = 1/9
35.(1) Total balls in the bag
= 6 + 4 + 8 = 18
Total possible outcomes = selection of3 balls out of18
balls=
18C = (18 × 17 × 16)/ (1 × 2 × 3) = 816
3
Favourable outcomes = Selection of 2 balls out of 6 red
balls and that of 1 ball out of 4 green balls
= 6C2 4C1 = 6 × 5/2× 4 = 60
∴ Required probability = 60/816 = 5/68
36.(2) Total possible outcomes = 18C1 = 18
Total possible outcomes = 18C1 = 18
Total favourable outcomes =selection of 1 ball out of 8
yellow balls = 8C1 = 8 = 4/9
∴ Required Probability = 8/18 = 4/9
37.(3) Total possible outcomes
= 18C2 = (18 × 17)/1 × 2 = 153
Total favourable outcomes
= 6C2 + 4C2 + 8C2
= (6 × 5)/(1 × 2) + (4 × 3)/(1 × 2) + (8 × 7)/(1 × 2)
= 15 + 6 + 28 = 49
Required probability = 49/153
38.(3) Total possible outcomes
= selection of 2 balls out of 12 balls
= 12C2 = (12 × 11)(1 × 2) = 66
Favourable outcomes = selection of 2 balls out 7 blue
balls
= 7C2 = 7 × 6/(1 × 2) = 21
∴ Required probability
= 21/66 = 7/22
39.(4) Total possible outcomes
= 6 × 6 = 36
Favourable outcomes
= (1,6) (6,1) (2,5) (5,2) (3,4) (4,3) = 6
∴ Required probability
= 6 / 36 = 1/6
Calculations (18 - 20)
males = 2/3 × 240 = 160
Females = 80
Graduate males= 15 × 160/100 = 24
Non-graduate males = 160 – 24 = 136
Graduate females = (80 × 3)/4 = 60
Non- graduate females = 20
40.(3) Required answer = 24 – 20 = 4
41.(5) Required answer = 60 + 136 = 196
42.(2) Required ratio
= 160 : 20 = 8 : 1
43.(1) Total balls in the bag
= 6 + 4 + 8 = 18
Total possible outcomes = selectionof 3 balls out of 18
balls 18C3 = (18 × 17 × 16)/ (1 × 2 × 3) = 816
Favorable outcomes = selection of 2 balls out of red balls
and that of 1 ball out of 4 green balls
= 6C2 × 4C1= (6 × 5)/2 × 4 = 60
∴ Required probability
= 60/816 = 5/68
44.(2) Total possible outcomes
= 18C1 =18
Total favourable outcomes = selection of 1 ball out of 8
yellow balls = 8C1 = 8
∴ Required probability = 8/18 = 4/9
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45.(3) Total possible outcomes
= 18C2 + 4C2 + 8C2
= (6 × 5)/(1 × 2) + (4 × 3)/(1 × 2) + (8 × 7)/(1 × 2)
= 15 + 6 + 28 = 49
Required probability = 49/153
46.(3) Total number of shirts in the bag = 4 + 5 + 6 = 15
Total possible outcomes Selectionof3 shirts out of15
shirts = 15C3
= (15 × 14 × 13)/(1 × 2 ×3) = 455
Favourable outcomes= selection of 1 blue shirt out of 4
blue shirts and selection of 2 shirts out of remaining 11
shirts
= 4C1 × 11C2 = 4 × (11 × 10)/2 = 220
∴ Required probability
= 220/455 = 44/91
47.(3) Total possible outcomes = 15C1= 15
Favourable outcomes = Selection of 1 red shirt or
selection of 1 yellow shirt.
= 5C1 + 6C1 = 5 + 5 = 11
∴ Required probability = 11/15
48.(3) Total possible outcomes = 15C2 = (15 × 14) /(1 ×
2) = 105
Favourable outcomes = Selection of 2 blue shirts out of 4
blue shirts
= 4C2 = (4×3)/(1×2) = 6
∴ Required probability = 6 / 105 = 2/35
49.(4) Total number of balls in the bag = 4 + 4 + 2 = 10
Total possible outcomes = Selection of 2 balls out of 10
balls
= 10C2 = (10 × 9)/ (1 × 2) = 45
Favourable outcomes = 1 green ball and 1 ball of other
colour +2 green balls
= 2C1 × 8C1 + 2C2= 2 × 8 × 2 = 18
∴ Required probability
= 18/45 = 2/5
50.(3) Total number of balls in the bag
= 5 + 6 + 3 = 14
Total possible outcomes
= Selection of 2 balls out of 14 balls
= 14C2= (14 × 13)/(1 × 2) = 91
Favourable outcomes = Selection of 2 balls out of 5 red
balls + selection of 2 balls out of 3 green balls = 5C2 + 3C2
= (5 × 4)/(1 × 2) + (3 × 2)/(1 × 2) = 10 + 3
= 13
∴ Required probability
= 13/91 = 1/9
51.(3) Total number of balls
= 4 + 6 + 5 = 15
Total possible outcomes = selection of 3 balls out of 15
balls
= 15C3 = (15 × 14 × 13)/ (1 × 2 × 3)
= 455
Total favourable outcomes = selection of 2 balls out of 6
green balls and selection of 1 ball out of 5 blue balls
= 6C2 × 5C2 = (6 × 5)/(1 × 2) ×5
= 75
∴ Required probability
= 75/455 = 15/91
52.(2) Total no. of cases = 7C2 = 21
Favourable no. of cases = 5C2 = 10
∴ Required probability = 10/21
53.(5) Total number of marbles in the urn = 15
P(S) = Total possible outcomes = Selection of 2 marbles
at random out of 15 marbles
= 15C2 = (15 × 14)/(1 × 2) = 105
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P(E) Favourable outcomes
= Selection of 2 marbles at random out of 2 green
marbles 2C2= 1
∴ Required probability
= P(E)/P(S) = 1/105
54.(3) P(S) = 15C3
= (15 × 14 × 13)/(1 × 2 × 3)
= 455
P(E) = Selection of 2 marbles out of 6 blue marbles and
that of one marbles out 4 yellow marbles
= 8C2 × 4C1 = (6 × 5) / (1 × 2) × 4 = 60
∴ Required probability
= P(E)/P(S) = 60/455 = 12/91
55.(2) P(S) = 15C4
= (15 × 14 × 13 × 12)/(1 × 2 × 3 × 4) = 1365
Let no yellow marble is selected.
∴ P(E) Selection of 4 marbles our of 11 marbles
= 11C4 = ( 11 × 10 × 9 × 8)/(1 × 2 × 3 × 4)= 300
∴ Required probability = 1 – 300/1365 = 1 – 22/91 =
91 – 22/91 = 69/91
56.(2) P(S) = 15C2 = 105
P(E) = 3C2 + 2C2
= (3 × 2)/(1 × 2) + 1 = 4
∴ Required probability = 4/105
57.(3) P(S) = 15C4 = 1365
P(E) = 2C1 × 6C2 × 3C1
= 2 × 15 × 3 = 90
∴ Required probability = P(S) / P (E)
= 90/1365 = 6/91
58.(3) Total possible outcomes
= 17C5 = (17 × 16 × 15 × 14 × 13)/(1 × 2× 3 × 4 ×5)
= 6188
Total favourable outcomes
= 8C2 ×4C1 ×5C2
= (8 × 7)/(1 ×2) × 4 (5 × 4)/(1 × 2)
= 28 × 4 × 10 = 1120
Required probability
= 1120/6188 = 280/1547
59.(3) Total balls in bag A
= 4 + 6 = 10
Green balls = 4
Probability that ball is green
= 4/10
Total balls in bag B = 7
Green balls = 3
= 3/7
∴ Total probability = 4/10 × 3 /7
= 12/70 = 6 / 35
60.(4) Out of 14 eggs, 8 eggs are rotten.
Hence, 6 eggs are not rotten
Total possible outcomes
= Selection of 2 eggs out of 14 eggs
= 14C2 = (14 × 13)/ (1 × 2) = 91
Total favourable outcomes = Selection of 2 eggs out of 6
fresh eggs
= 6C2 = (6 × 5)/(1 × 2) = 15
∴ Required probability = 15/91
61.(3) Total balls in bag A
= 4 + 6 = 10
Green balls = 4
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Probability that ball is green = 4/10
Total balls in bag B = 7
Green balls = 3
Probability that ball is green = 3/7
∴ Total probability = 4/10 × 3/7
= 12/70 × 6/35
62.(4) Total possible outcomes
= 13C2 + 7C2
= (13 × 12)/(1 × 2) + (7 × 6)/(1 × 2)
= 78 + 21 = 99
∴ Required probability = 99/190
63.(3) Total balls = 4 + 3 + 5 = 12
Out of 12, we are drawing 2 balls at random
The 2 balls can be drawn in 12C2 ways. Then probability
both balls are of same colour
= 4C2 + 3C2 + 5C2/12C2 = (3 + 6 + 10)/(6 × 11) = 19/66
64.(3) Three balls are drawn.
∴ Total possible outcomes
= 18C3 = (18 × 17 × 16)(1 × 2 × 3) = 816
Favourable outcomes – Selection of two balls out of 5 red
balls and that of 1 ball from 7 blue balls
= 5C2 × 7 C1 = 10 × 7 = 70
∴ Required probability = 70/816 = 35/408
65.(2) n(S) = Numberof ways of selecting 3 marbles out
of 7 marbles = 7C3
= (7 × 6 × 5)/(1 × 2 ×3) = 35
n (E) = Number of ways of selecting 2 green marbles out
1 red marbles
= 4C2 × 3C1 = (4 × 3)/(1 ×2) × 3 = 18
∴ Required probability
= n(E)/n(S) = 52/100 = 0.52
66.(2) Favourable number of cases
= (40 × 70)/100 + (30 × 80)/100
= 28 + 24 = 52
∴ Required probability = 52/100 = 0.52
67.(1) Total possible outcomes
= Selection of 2 eggs pit of 28
= 28C2 = 28 × 27/(1 × 2) = 14 × 27
Favourable outcomes
= Selection 1 unrotten egg out 20 and 1 rotten egg out
of 8 + selecting 2 rotten eggs out 8 = 20C1 × 8C1 + 8C2
= 160 + 8 × 7/2 = 188
∴ Required probability
= 188/14 × 27 = 94/7 × 27 = 94/189
68.(2) Required probability = 3/8
[∴ Total sector = 8
Sectors having even No. = 3]
69.(1) Required probability = 1/5
70.(1) Total number of balls in the urn = 5 + 6 + 7 = 18
Two balls are drawn.
∴ Total possible outcomes
= 18C2= (18 × 17)/ (1 × 2) = 153
Favourable outcomes
= 5C2 = 5 × 4/(1 × 2) = 10
∴ Required probability = 10/153
71.(2) Total possible outcomes
= 18C2 = (18 × 17)/(1 × 2) = 153
Facourable outcomes = 6C2
= (6 × 5) / (1 ×2) = 15
∴ Required Probability = 15/153
= 5/51
72.(5) Total number of balls in the box = 4 + 4 + 6 = 14
Total possible outcomes
= Selection of 2 balls out of 14 balls = 14C2= (14 × 13)/(1
× 2) = 91
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Total favourable outcomes = Selction of 1 ball out of 6
blue balls and other ball out 8 remaining balls +
Selection of 2 balls out of 6 blue balls
= 6C1 + 8C1 + 6C2
= 6 × 8 + (6 × 5)(1 × 2) = 48 + 15 = 63
∴ Required probability = 63/91
= 9/13
73.(5) Total possible outcomes = 14C1 = 14
Total favourable outcomes
= 4C1 + 6C1 = 4 + 6 = 10
∴ Required probability
= 10/14 = 5/7
74.(4) Total possible outcomes
= 14C3 = (14 × 13 × 12)/(1 × 2 × 3) = 364
Total favourable outcomes
= 4C1 × 10C1 = 4 × 10 = 40
∴ Required probability = 40/364 = 10/91
75.(1) There are 10 balls in the bag.
Total possible outcomes = Selection of 2 balls out of 10
balls
= 10C2 = (10 × 9) (1 × 2) = 45
Total favourable outcomes
= Selection of 2 balls out of 6 red balls + selection of 2
balls out of 4 yellow balls
= 6C2 + 4C2
= (6 × 5)/(1 × 2) + (4 × 3)/(1 × 2)
= 15 + 6 = 21
∴ Required probability
= 21/45 = 7/15
76.(5) Total balls in the bag
= 7 + 8 = 15
Total possible outcomes = Selection of 2 balls out of 15
balls
= 15C2 = (15 × 14) / (1 × 2) = 105
Total favourable outcomes = Selection of 2 balls out of
yellow balls + Selection of 2 balls out of 7 red balls
= 8C2 + 7C2 = (8 × 7)/ (1 × 2) + (7 × 6)/(1 × 2)
= 28 + 21 = 49
Required probability
= 49/105 = 7/15
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